Let $(5, 6)$ and $(9, - 4)$ be two points on a parabola . The point of intersection of tangents at these points is $(1, - 2) .$ Then the slope of directrix of the parabola is

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Solution

The correct option is D $- 2$
Let $A (a t_{1}^{2}, 2 a t_{1})$ and $B (a t_{2}^{2}, 2 a t_{2})$ be two points on the parabola $y^{2} = 4 a x .$ The point of intersection of tangents at these points is $P (a t_{1} t_{2}, a (t_{1} + t_{2}))$ and the mid-point of $A B$ is $M (\frac{a (t_{1}^{2} + t_{2}^{2})}{2}, a (t_{1} + t_{2}))$
$\Rightarrow M P ⊥ Directrix$

So, the line joining the mid-point of any two points on the parabola and point of intersection of tangents at this point, is perpendicular tangent.

Two given points are $(5, 6)$ and $(9, - 4) .$ The midpoint of them is $(7, 1) .$ The slope of line joining $(7, 1)$ and $(1, - 2)$ (point of intersection of tangents) is $\frac{1}{2} .$
So, the slope of directrix is $- 2.$

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